Coloring the Cartesian sum of graphs

For graphs G and H, let [email protected]?H denote their Cartesian sum. We investigate the chromatic number and the circular chromatic number for [email protected]?H. It has been proved that for any graphs G and H, @g([email protected]?H)@?max{@[email protected]"c(G)@g(H)@?,@[email protected](G)@g"c(H)@?}. It has been conjectured that for any graphs G and H, @g"c([email protected]?H)@?max{@g(H)@g"c(G),@g(G)@g"c(H)}. We confirm this conjecture for graphs G and H with special values of @g"c(G) and @g"c(H). These results improve previously known bounds on the corresponding coloring parameters for the Cartesian sum of graphs.

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